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%	TITLE SECTION 
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\title{Quadratic Function} % Poster title

\author{matematika.pl} % Author(s)

\institute{2015} % Institution(s)

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\begin{document}

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%	OBJECTIVES
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\begin{alertblock}{Objectives}
Objectives for today:
\begin{itemize}
\item 介绍一下你自己。
\item Introducing specific vocabulary.
\item Quick revision of quadratic function.
\item Factorising Quadratics.
\item Proving Vieta's formulas.
\item Carrying out gained knowledge by working out some word problems.
\end{itemize}

\end{alertblock}

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%	QUICK REVISION
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\begin{block}{Quick Revision}

\textbf{Forms of Quadratic Function}
\begin{itemize}
\item $f(x) = ax^2+bx+c$ is called the \textbf{standard form}.
\item $f(x) = a(x-x_1)(x-x_2)$ is called the \textbf{factored form}, where $x_1$ and $x_2$ are the roots of the quadratic function.
\item $f(x) = a(x-h)^2+k$ is called the \textbf{vertex form}.
\end{itemize}

\textbf{Delta $\Delta$}\\*
$\Delta$ determines tells us how many solutions quadratic equation have:
$$\text{number of solutions}=
\begin{cases}
2 &\text{when } \Delta > 0\\
1 &\text{when } \Delta = 0\\
0 &\text{when } \Delta < 0
\end{cases}
$$


\textbf{The Quadratic Formula}
$$x = \frac{-b\pm \sqrt{\Delta}}{2a}$$

\textbf{Graph of Quadratic Function}

\end{block}

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\begin{figure}
\includegraphics[width=0.8\linewidth]{1.jpg}
\caption{Graph of $f(x)=ax^2|_{\{0.1, 0.3, 1.0, 3.0\}}$}
\end{figure}

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%	MATERIALS
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\begin{block}{Factorising a Quadratic}

Factorising a quadratic means putting it into two brackets, and is useful if you're trying to draw a graph of a quadratic solve a quadratic equation. It's pretty easy if $a=1$ (in $ax^2+bx+c$ form), but can be a real pain otherwise.
\newline
\newline
In order to factorise a quadratic you should follow steps outlined below:

\begin{enumerate}
\item Rearrange the equation into the standard $ax^2+bx+c$ form.
\item Write down two brackets: $(x\ \ \ )(x\ \ \ )$
\item Find two numbers that multiply to give 'c' and add or subtract to give 'b' (ignoring signs).
\item Put the numbers in brackets and choose their signs.
\end{enumerate}

\end{block}

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%	P
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\begin{block}{Factorising- Tasks}
1. Factorise $x^2-x-12$.
\[\]
\[\]
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\[\]
2. Solve $x^2-8=2x$ by factorising.

\end{block}

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%	IMPORTANT To REMEMBER
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\begin{alertblock}{Myth of Delta $\Delta$}

It's commonly believed that in order to work out roots of a quadratic function you must count $\Delta$ and use other previously established formulas. However this is untrue since factorising in many cases is as good or even better than simply counting $\Delta$.

\end{alertblock} 

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%	EXAMPLE OF FACTORISATION
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\begin{block}{Example of Factorisation}
Solve $x^2+4x-21=0$ by factorising.

$$x^2+4x-21=(x\ \ \ \ \ )(x\ \ \ \ \ )$$
$1$ and $21$ multiply to give $21$ - and add or subtract to give $22$ and $20$.\\*
$3$ and $7$ multiply to give $21$ - and add or subtract to give $10$ and \textbf{$4$}.

$$x^2+4x+21 = (x+7)(x-3)$$

And solving the equation:
$$(x+7)(x-3)=0$$
we get
$$x=-7,\ \ \ x=3$$

\end{block}

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%	PROOF OF VIETA'S FORMULAS
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\begin{block}{ Proof of Vieta's Formulas}
Let's prove that:
$$x_1 + x_2 = \frac{-b}{a}$$
When $\Delta$ is positive we have two roots:
$$x_1 = \frac{-b-\sqrt{\Delta}}{2a},\ \ \ x_2 = \frac{-b+\sqrt{\Delta}}{2a}$$

Substituting for $x_1$ and $x_2$ respectively, we receive:
					
$$x_1 + x_2 = \frac{-b-\sqrt{\Delta}}{2a} + \frac{-b+\sqrt{\Delta}}{2a} =$$
$$ = \frac{(-b-\sqrt{\Delta}) + (-b+\sqrt{\Delta})}{2a} = \frac{-2b}{2a} = \frac{-b}{a}$$
                    
The same we could do with another pattern, which state that $x_1 x_2 = \frac{c}{a}$, but proving this is going to be your task in next section.

\end{block}

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%	CONCLUSION
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\begin{block}{Vieta's Formulas- Task}
1. Prove that $$x_1x_2 = \frac{c}{a}$$
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\end{block}


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%	ACKNOWLEDGEMENTS
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\begin{block}{Glossary}

\begin{table}
\vspace{2ex}
\begin{tabular}{l l l l}
\toprule
\textbf{verb} & \textbf{noun} & \textbf{meaning}\\
\midrule
add & addition & $+$ \\
subtract & subtraction & $-$ \\
multiply & multiplication & $\cdot$ \\
divide & division & $\div$ \\
solve & solution & getting answer \\
substitute & substitution & $t=x^2$ \\



\bottomrule
\end{tabular}
\caption{Word Formation}
\end{table}


\end{block}

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\begin{alertblock}{Some Necessary and Useful Vocabulary}

\begin{itemize}
\item (n.) sign $\rightarrow$ $+$ or $-$
\item (n.) equation $\rightarrow something = 0$ 
\item (n.) factor $\rightarrow$ two multiplied factors give result
\item (v.) factorise $\rightarrow$ putting into brackets
\item (n.) coefficient $\rightarrow$ a constant number i.e. $a$, $b$, $c$ in a pattern $ax^2+bx+c$
\item (n.) quadratic function $\rightarrow$ $f(x) = ax^2+bx+c$
\item (n.) root $\rightarrow$ $\sqrt{sth}$ or solution of quadratic equation
\item (n.) formula $=$ pattern
\end{itemize}

\end{alertblock}


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\end{document}
